Fractional Calculus and Applied Analysis最新文献

Abstract

This work explores the possibility of developing the analog of some classic results from elliptic PDEs for a class of fractional ODEs involving the composition of both left- and right-sided Riemann-Liouville (R-L) fractional derivatives, ({D^alpha _{a+}}{D^beta _{b-}}) , (1<alpha +beta <2) . Compared to one-sided non-local R-L derivatives, these composite operators are completely non-local, which means that the evaluation of ({D^alpha _{a+}}{D^beta _{b-}}u(x)) at a point x will have to retrieve the information not only to the left of x all the way to the left boundary but also to the right up to the right boundary, simultaneously. Therefore, only limited tools can be applied to such a situation, which is the most challenging part of the work. To overcome this, we do the analysis from a non-traditional perspective and eventually establish elliptic-type results, including Hopf’s Lemma and maximum principles. As (alpha rightarrow 1^-) or (alpha ,beta rightarrow 1^-) , those operators reduce to the one-sided fractional diffusion operator and the classic diffusion operator, respectively. For these reasons, we still refer to them as “elliptic diffusion operators", however, without any physical interpretation.

In this paper, the non-autonomous fractional evolution inclusions of Clarke subdifferential type in a separable reflexive Banach space are investigated. The mild solution of the non-autonomous fractional evolution inclusions of Clarke subdifferential type is defined by introducing the operators (psi (t,tau )) and (phi (t,tau )) and V(t), which are generated by the operator (-mathcal {A}(t)) and probability density function. Combined the measure of non-compactness, some properties of the Clarke subdifferential with fixed point theorem of (kappa -)condensing multi-valued maps, a new existence result of mild solution is established. Moreover, an existence result of optimal control pair for the Lagrange problem is also derived. The results obtained in this paper extend the study of fractional autonomous evolution equations to the non-autonomous fractional evolution inclusions. Finally, a fractional partial differential inclusion with control is provided to illustrate the applications of the obtained main results.

Abstract

In this work we introduce discrete convolution operators and study their most basic properties. We then solve linear difference equations depending on such operators. The theory herein developed generalizes, in particular, the theory of discrete fractional calculus and fractional difference equations. To that matter we make use of the so-called Sonine pairs of kernels.

We study the approximate optimal control for a class of fractional stochastic hemivariational inequalities with non-instantaneous impulses driven by Rosenblatt process in a Hilbert space. Firstly, a suitable definition of piecewise continuous mild solution is introduced, and by using stochastic analysis, properties of (alpha )-order sine and cosine family and Picard type approximate sequences, we show the existence and uniqueness of approximate mild solutions for the inequality problems of fractional order (1, 2] under the non-Lipschitz conditions. Secondly, we provide the existence conditions of approximate solutions to optimal control problems driven by the presented control systems with the help of a new minimizing sequence method. Finally, an example is provided to illustrate the theory.

Abstract

In this paper we prove existence of solutions to Schrödinger-Maxwell type systems involving mixed local-nonlocal operators. Two different models are considered: classical Schrödinger-Maxwell equations and Schrödinger-Maxwell equations with a coercive potential, and the main novelty is that the nonlocal part of the operator is allowed to be nonpositive definite according to a real parameter. We then provide a range of parameter values to ensure the existence of solitary standing waves, obtained as Mountain Pass critical points for the associated energy functionals.

Abstract

In this article, we focus on the application of the recent notion of time-fractional derivative developed in Sobolev spaces to the study of well-posedness and stability for a time-fractional wave-like equation with superdiffusion and subdiffusion terms.

Abstract

In this work we study the Riemann-Liouville fractional integral of order (alpha in (0,1/p)) as an operator from (L^p(I;X)) into (L^{q}(I;X)) , with (1le qle p/(1-palpha )) , whether (I=[t_0,t_1]) or (I=[t_0,infty )) and X is a Banach space. Our main result provides necessary and sufficient conditions to ensure the compactness of the Riemann-Liouville fractional integral from (L^p(t_0,t_1;X)) into (L^{q}(t_0,t_1;X)) , when (1le q< p/(1-palpha )) .

Let (0<qle infty ), b be a slowly varying function and ( Phi : [0,infty ) longrightarrow [0,infty ) ) be an increasing function with (Phi (0)=0) and (lim limits _{r rightarrow infty }Phi (r)=infty ). In this paper, we introduce a new class of function spaces (L_{Phi ,q,b}) which unify and generalize the Lorentz-Karamata spaces with (Phi (t)=t^p) and the Orlicz-Lorentz spaces with (bequiv 1). Based on the new spaces, we introduce five new Hardy spaces containing martingales, the so-called Orlicz-Lorentz-Karamata Hardy martingale spaces and then develop a theory of these martingale Hardy spaces. To be precise, we first investigate several properties of Orlicz-Lorentz-Karamata spaces and then present Doob’s maximal inequalities by using Hardy’s inequalities. The characterization of these Hardy martingale spaces are constructed via the atomic decompositions. As applications of the atomic decompositions, martingale inequalities and the relation of the different martingale Hardy spaces are presented. The dual theorems and a new John-Nirenberg type inequality for the new framework are also established. Moreover, we study the boundedness of fractional integral operators on Orlicz-Lorentz-Karamata Hardy martingale spaces. The results obtained here generalize the previous results for Lorentz-Karamata Hardy martingale spaces as well as for Orlicz-Lorentz Hardy martingales spaces. Especially, we remove the condition that b is non-decreasing as in [38, 39] and the condition (q_{Phi ^{-1}}<1/q) in [24], respectively.

This paper presents a numerical method based on an operational matrix of Legendre polynomials for resolving the class of time fractional diffusion (TFD) equations. The operational matrix of fractional order derivatives of the Legendre polynomials is derived as a product of matrices. The collocation method together with the operational matrix of Legendre polynomials are employed to transform the TFD equations into a set of algebraic equations. The perturbation method is applied to show the stability of the discussed method. The accuracy of the suggested method is validated using numerical experiments. The solution obtained by this method is in excellent agreement with the exact solution for the integer order of derivatives and is more precise than the solution obtained by the existing method in which Bernstein polynomials are taken as the basis polynomials.

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis–Nirenberg problem.